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Section 11.5 Partial Fraction Decomposition Copyright © 2013 Pearson Education, Inc. All rights reserved

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$Page 1: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

Section 11.5

Partial Fraction Decomposition

Copyright © 2013 Pearson Education, Inc. All rights reserved

$Page 2: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

$Page 3: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

$Page 4: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

$Page 5: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

2

7 1Write the partial fraction decomposition of 6

xx x

+

+ −( 3)( 2)x x+ −

2

7 16 3 2

x A Bx x x x

+= +

+ − + −

( )2

( 3)( 2) ( 3)( 2) ( 37 1 )(6

23 2

)x x x x x xx A Bx x x x

+ − + −+=

+

+ −+

+ − −

7 1 ( 2) ( 3)x A x B x+ = − + +

( 3)B x +

( ) ( )2 3A B x A B= + + − +

Equate coefficients of like powers of x to get: 71 2 3

A BA B

= +⎧⎨= − +⎩

4 3A B= =

2

7 1 4 36 3 2

xx x x x

+= +

+ − + −

$Page 6: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

$Page 7: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

$Page 8: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

( )( )

2

22Write the partial fraction decomposition of

1 2x

x x+

− +

( )( ) ( )

2

2 22

1 21 2 2x A B C

x xx x x+

= + +− +− + +

( )( )2Clear of fractions by multiplying all terms by 1 2x x− +

( ) ( )( ) ( )22 2 2 1 2 1x A x B x x C x+ = + + − + + −

2 2 22 4 4 2x Ax Ax A Bx Bx B Cx C+ = + + + + − + −

( ) ( ) ( )2 22 4 4 2x A B x A B C x A B C+ = + + + + + − −

10 42 4 2

A BA B CA B C

= += + += − −

1 2, , 23 3

A B C= = = −

( )2

1 223 3

1 2 2x x x−

= + +− + +

$Page 9: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

( )

2

23

4Write the partial fraction decomposition of 1

xx x

−

−

( ) ( )

2

2 22 33

411 1

x A B C D Ex x x xx x x

−= + + + +

−− −

( ) ( ) ( ) ( )2 2 22 2 3 34 1 1 1 1x Ax x Bx x C x Dx x Ex− = − + − + − + − +

( )2Let 0: 4 0 1x C= − = − 4C = − ( )32Let 1: 1 4 1x E= − = 3E = −

( ) ( ) ( ) ( )2 2 22 3 2 34 4 1 3 1 1 1x x x Ax x Bx x Dx x− + − + = − + − + −

( ) ( ) ( ) ( )( )23 8 ( 1) 1 1 1x x x x x Ax x B x Dx+ − = − − + − +

( ) ( ) 23 8 1 1x Ax x B x Dx+ = − + − + ( )Let 0 :8 0 1x B= = − 8B = −

( )2Let 1:3(1) 8 1x D= + = 11D =

( ) ( ) 23 8 1 8 1 11x Ax x x x+ = − − − +

( ) ( )( ) ( ) ( )2Let 2 :3 2 8 2 2 1 8 2 1 11 2x A= + = − − − + 11A = −

( ) ( )

2

2 22 33

4 11 8 4 11 311 1

xx x x xx x x

− − − − −= + + + +

−− −

$Page 10: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

$Page 11: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

$Page 12: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

( )( )2

1Write the partial fraction decomposition of 1 4x x+ +

( )( ) 22

11 41 4A Bx Cx xx x

+= +

+ ++ +

( ) ( )( )21 4 1A x Bx C x= + + + +

2 21 4Ax A Bx Bx Cx C= + + + + +

( ) ( ) ( )21 4A B x B C x A C= + + + + +

001 4

A BB CA C

= += += +

151515

A

B

C

=

= −

=

2

1 1 15 5 51 4

x

x x

− += +

+ +

$Page 13: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

$Page 14: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

$Page 15: Section 11.5 Partial Fraction Decomposition 2 2 Write the partial fraction decomposition of 12 x xx + −+ ( )( ) ( ) 2 22 2 12 212 xABC xx xxx + =+ + −+ +−+ Clear of fractions$

( )

2

22

1Write the partial fraction decomposition of 4

x

x

+

+

( ) ( )

2

2 222 2

144 4

x Ax B Cx Dxx x

+ + += +

++ +

( )( )2 21 4x Ax B x Cx D+ = + + + +

2 3 21 4 4x Ax Bx Ax B Cx D+ = + + + + +

( ) ( )2 3 21 4 4x Ax Bx A C x B D+ = + + + + +

010 41 4

ABA CB D

=== += +

0103

ABCD

==== −

( )22 2

1 34 4x x

−= +

+ +

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